Questions and Answers
What is the factorization of $72x^7y^2 - 16x^2y^3$?
What is the factorization of $-9y^9 - 18y^9x^2$?
What is the factorization of $20a^4b^5 + 8a^5b - 4a^3b$?
What is the factorization of $x^4 + 3x^2 - 7x$?
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What is the factorization of $x^3 + 3x^2 + 4x + 12$?
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What is the factorization of $25xy + 30x - 30y - 36$?
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What does Prime mean in factoring context?
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What is the factorization of $16x^3 + 8x^2 + 8x + 4$?
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What is the factorization of $n^2 - 8n + 7$?
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What is the factorization of $x^2 - 6x + 4$?
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What is the factorization of $n^2 - 2n - 35$?
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What is the factorization of $3x^2 - 15x + 12$?
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What is the factorization of $6x^2 - 42x - 108$?
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What is the factorization of $5x^2 + 27x + 10$?
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What is the factorization of $5v^2 - 23v + 12$?
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What is the factorization of $6k^2 - 3k - 9$?
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What is the factorization of $10n^2 - 26n + 12$?
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What is the factorization of $25m^2 - 1$?
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What is the factorization of $16r^2 + 8r + 1$?
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What is the factorization of $9k^2 - 12k + 4$?
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What does Prime mean in the context of polynomials?
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What is the factorization of $4x² + 13x + 9$?
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What is the factorization of $3n^2 - 75$?
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What is the factorization of $100b^2 + 120b + 36$?
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What is the factorization of $18a² - 24a + 8$?
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What is the factorization of $2x³ + 7x² - 2x - 7$?
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What is the factorization of $12x³ - 20x² - 3x + 5$?
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Study Notes
Factoring Basics
- Factoring involves rewriting expressions as the product of their factors.
- The greatest common factor (GCF) is the highest factor shared among terms.
Types of Factorization
- GCF Only: Some expressions can be factored by identifying a common factor among all terms.
- Factor by Grouping: Typically used for polynomials that can be grouped into pairs with common factors.
- Trinomials: Quadratic expressions can often be factored into two binomials, especially when the leading coefficient (LC) is 1.
- Special Cases: Certain forms, such as perfect square trinomials and differences of squares, follow specific factoring patterns.
Examples of Factoring
-
GCF Examples:
- ( 8x²y²(9x⁵ - 2y) ) becomes ( 72x⁷y² - 16x²y³ ).
- ( -9y⁹(1 + 2x²) ) leads to ( -9y⁹ - 18y⁹x² ).
-
Trinomial Factoring:
- ( n² - 8n + 7 ) is factored using ( (n - 7)(n - 1) ).
- ( n² - 2n - 35 ) factors to ( (n + 5)(n - 7) ).
-
Special Cases:
- The expression ( (5m + 1)(5m - 1) ) applies the difference of squares to yield ( 25m² - 1 ).
- ( (4r + 1)² ) expands to ( 16r² + 8r + 1 ) (perfect square trinomial).
Identifying Prime Expressions
- Certain expressions cannot be factored further, labeled as "Prime," like ( x² - 6x + 4 ) where no pairs of numbers multiply to 4 and add to -6.
Application of GCF and Special Cases
- Problems often combine GCF with special cases: ( 3(n + 5)(n - 5) ) simplifies to ( 3n² - 75 ) combining the difference of squares with GCF.
Summary of Key Techniques
- Recognize the structure of the polynomial to determine the best factoring method.
- Different methods may apply to the same polynomial depending on its form, such as grouping, recognizing trinomials, or applying GCF first.
- Practicing various factoring methods helps in identifying the quickest path to the solution for polynomials.
Studying That Suits You
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Description
Test your skills in factoring expressions with these flashcards designed for Algebra 1. Each card presents a problem to factor completely, focusing on the Greatest Common Factor (GCF). Use these cards to solidify your understanding and improve your algebra skills.